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Properties

Label 1062.2.e.d
Level $1062$
Weight $2$
Character orbit 1062.e
Analytic conductor $8.480$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1062,2,Mod(355,1062)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1062, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1062.355");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1062 = 2 \cdot 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1062.e (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.48011269466\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 19x^{12} + 118x^{10} + 306x^{8} + 363x^{6} + 198x^{4} + 45x^{2} + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{2} - \beta_1 q^{3} + \beta_{3} q^{4} + (\beta_{7} - \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{5}+ \cdots + (\beta_{8} - \beta_{7} + \beta_{6} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{2} - \beta_1 q^{3} + \beta_{3} q^{4} + (\beta_{7} - \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{5}+ \cdots + (2 \beta_{13} + \beta_{12} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 7 q^{2} + q^{3} - 7 q^{4} + 4 q^{5} + 5 q^{6} - 14 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 7 q^{2} + q^{3} - 7 q^{4} + 4 q^{5} + 5 q^{6} - 14 q^{8} + 5 q^{9} + 8 q^{10} - 7 q^{11} + 4 q^{12} + 9 q^{13} + 5 q^{15} - 7 q^{16} + 28 q^{17} + 4 q^{18} - 22 q^{19} + 4 q^{20} - 9 q^{21} + 7 q^{22} - 17 q^{23} - q^{24} - 5 q^{25} + 18 q^{26} - 2 q^{27} + 7 q^{29} - 14 q^{30} + 13 q^{31} + 7 q^{32} - 14 q^{33} + 14 q^{34} - 24 q^{35} - q^{36} - 38 q^{37} - 11 q^{38} + 9 q^{39} - 4 q^{40} - 23 q^{41} + 12 q^{42} - 3 q^{43} + 14 q^{44} + 49 q^{45} - 34 q^{46} + q^{47} - 5 q^{48} + 7 q^{49} + 5 q^{50} + 23 q^{51} + 9 q^{52} + 72 q^{53} - 10 q^{54} - 32 q^{55} - 23 q^{57} - 7 q^{58} - 7 q^{59} - 19 q^{60} + 2 q^{61} + 26 q^{62} + 36 q^{63} + 14 q^{64} - 30 q^{65} + 8 q^{66} + 8 q^{67} - 14 q^{68} + 2 q^{69} - 12 q^{70} + 60 q^{71} - 5 q^{72} - 10 q^{73} - 19 q^{74} - 37 q^{75} + 11 q^{76} - 42 q^{77} - 12 q^{78} - 11 q^{79} - 8 q^{80} - 19 q^{81} - 46 q^{82} - 19 q^{83} + 21 q^{84} + 35 q^{85} + 3 q^{86} - 16 q^{87} + 7 q^{88} - 30 q^{89} + 17 q^{90} + 12 q^{91} - 17 q^{92} - 7 q^{93} - q^{94} + q^{95} - 4 q^{96} + 4 q^{97} + 14 q^{98} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 19x^{12} + 118x^{10} + 306x^{8} + 363x^{6} + 198x^{4} + 45x^{2} + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 121 \nu^{13} - 133 \nu^{12} - 2321 \nu^{11} - 2372 \nu^{10} - 14700 \nu^{9} - 12900 \nu^{8} + \cdots - 2733 ) / 1314 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 61\nu^{12} + 1190\nu^{10} + 7713\nu^{8} + 21024\nu^{6} + 24771\nu^{4} + 10629\nu^{2} + 855 ) / 219 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 586\nu^{13} + 10703\nu^{11} + 61239\nu^{9} + 133590\nu^{7} + 110445\nu^{5} + 25494\nu^{3} - 1785\nu - 657 ) / 1314 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 421 \nu^{13} + 431 \nu^{12} + 7538 \nu^{11} + 7909 \nu^{10} + 41313 \nu^{9} + 45726 \nu^{8} + \cdots + 3729 ) / 1314 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 332 \nu^{13} + 465 \nu^{12} - 6010 \nu^{11} + 8382 \nu^{10} - 33639 \nu^{9} + 46539 \nu^{8} + \cdots - 4059 ) / 1314 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 332 \nu^{13} - 707 \nu^{12} - 6010 \nu^{11} - 13024 \nu^{10} - 33639 \nu^{9} - 75939 \nu^{8} + \cdots - 3117 ) / 1314 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 121 \nu^{13} - 971 \nu^{12} + 2321 \nu^{11} - 18088 \nu^{10} + 14700 \nu^{9} - 107952 \nu^{8} + \cdots - 11901 ) / 1314 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1073 \nu^{13} + 183 \nu^{12} + 20164 \nu^{11} + 3570 \nu^{10} + 122217 \nu^{9} + 23139 \nu^{8} + \cdots + 2565 ) / 1314 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1076 \nu^{13} - 604 \nu^{12} + 19684 \nu^{11} - 11108 \nu^{10} + 113226 \nu^{9} - 64452 \nu^{8} + \cdots - 5145 ) / 1314 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1448 \nu^{13} + 350 \nu^{12} + 27178 \nu^{11} + 6415 \nu^{10} + 164517 \nu^{9} + 36852 \nu^{8} + \cdots - 1176 ) / 1314 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1529 \nu^{13} + 288 \nu^{12} + 28672 \nu^{11} + 5166 \nu^{10} + 173391 \nu^{9} + 28413 \nu^{8} + \cdots - 2781 ) / 1314 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1702 \nu^{13} - 115 \nu^{12} - 31871 \nu^{11} - 1967 \nu^{10} - 192117 \nu^{9} - 9687 \nu^{8} + \cdots + 3540 ) / 1314 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 121 \nu^{13} - 1913 \nu^{12} - 2321 \nu^{11} - 35560 \nu^{10} - 14700 \nu^{9} - 211056 \nu^{8} + \cdots - 16725 ) / 1314 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{13} + 2\beta_{10} - 2\beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - 2\beta_{3} + \beta_{2} - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{13} + 2 \beta_{12} - \beta_{11} + 3 \beta_{10} + \beta_{9} - 2 \beta_{7} + 3 \beta_{6} + \cdots - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7 \beta_{13} + 3 \beta_{12} + \beta_{11} - 14 \beta_{10} + 3 \beta_{9} + 18 \beta_{8} + \cdots - 9 \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 20 \beta_{13} - 16 \beta_{12} + 8 \beta_{11} - 24 \beta_{10} - 8 \beta_{9} + 7 \beta_{7} - 30 \beta_{6} + \cdots + 41 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 64 \beta_{13} - 26 \beta_{12} + \beta_{11} + 117 \beta_{10} - 37 \beta_{9} - 162 \beta_{8} - 91 \beta_{7} + \cdots + 31 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 253 \beta_{13} + 134 \beta_{12} - 52 \beta_{11} + 186 \beta_{10} + 52 \beta_{9} + 19 \beta_{7} + \cdots - 313 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 623 \beta_{13} + 205 \beta_{12} - 71 \beta_{11} - 1074 \beta_{10} + 377 \beta_{9} + 1512 \beta_{8} + \cdots - 425 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2759 \beta_{13} - 1207 \beta_{12} + 350 \beta_{11} - 1557 \beta_{10} - 350 \beta_{9} - 653 \beta_{7} + \cdots + 2684 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 6103 \beta_{13} - 1700 \beta_{12} + 1003 \beta_{11} + 10203 \beta_{10} - 3703 \beta_{9} - 14418 \beta_{8} + \cdots + 4690 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 28327 \beta_{13} + 11309 \beta_{12} - 2590 \beta_{11} + 13899 \beta_{10} + 2590 \beta_{9} + \cdots - 24454 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 59678 \beta_{13} + 14986 \beta_{12} - 11312 \beta_{11} - 98238 \beta_{10} + 36104 \beta_{9} + \cdots - 48269 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 283004 \beta_{13} - 108037 \beta_{12} + 21119 \beta_{11} - 129156 \beta_{10} - 21119 \beta_{9} + \cdots + 230087 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 582475 \beta_{13} - 138020 \beta_{12} + 117745 \beta_{11} + 951423 \beta_{10} - 351547 \beta_{9} + \cdots + 482533 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1062\mathbb{Z}\right)^\times\).

\(n\) \(119\) \(415\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
355.1
− 1.45340i
− 1.06595i
− 2.21001i
3.12048i
0.810814i
− 0.601676i
− 0.332308i
1.45340i
1.06595i
2.21001i
− 3.12048i
− 0.810814i
0.601676i
0.332308i
0.500000 + 0.866025i −1.70983 − 0.276560i −0.500000 + 0.866025i 0.220438 − 0.381811i −0.615406 − 1.61904i 0.372327 + 0.644890i −1.00000 2.84703 + 0.945741i 0.440877
355.2 0.500000 + 0.866025i −1.19418 + 1.25456i −0.500000 + 0.866025i 0.980652 − 1.69854i −1.68357 − 0.406912i 1.42876 + 2.47469i −1.00000 −0.147852 − 2.99635i 1.96130
355.3 0.500000 + 0.866025i −1.13206 − 1.31089i −0.500000 + 0.866025i 1.65471 − 2.86604i 0.569238 − 1.63584i 0.457186 + 0.791869i −1.00000 −0.436885 + 2.96802i 3.30941
355.4 0.500000 + 0.866025i 0.427750 − 1.67840i −0.500000 + 0.866025i −0.0733879 + 0.127112i 1.66741 − 0.468758i −1.40439 − 2.43247i −1.00000 −2.63406 − 1.43587i −0.146776
355.5 0.500000 + 0.866025i 0.948586 − 1.44920i −0.500000 + 0.866025i −2.15423 + 3.73123i 1.72934 + 0.0968986i 1.22187 + 2.11634i −1.00000 −1.20037 − 2.74938i −4.30846
355.6 0.500000 + 0.866025i 1.43137 + 0.975290i −0.500000 + 0.866025i 1.26238 − 2.18651i −0.128943 + 1.72724i −2.15398 − 3.73080i −1.00000 1.09762 + 2.79199i 2.52477
355.7 0.500000 + 0.866025i 1.72837 − 0.112872i −0.500000 + 0.866025i 0.109436 − 0.189548i 0.961934 + 1.44038i 0.0782174 + 0.135476i −1.00000 2.97452 − 0.390169i 0.218871
709.1 0.500000 − 0.866025i −1.70983 + 0.276560i −0.500000 − 0.866025i 0.220438 + 0.381811i −0.615406 + 1.61904i 0.372327 − 0.644890i −1.00000 2.84703 − 0.945741i 0.440877
709.2 0.500000 − 0.866025i −1.19418 − 1.25456i −0.500000 − 0.866025i 0.980652 + 1.69854i −1.68357 + 0.406912i 1.42876 − 2.47469i −1.00000 −0.147852 + 2.99635i 1.96130
709.3 0.500000 − 0.866025i −1.13206 + 1.31089i −0.500000 − 0.866025i 1.65471 + 2.86604i 0.569238 + 1.63584i 0.457186 − 0.791869i −1.00000 −0.436885 − 2.96802i 3.30941
709.4 0.500000 − 0.866025i 0.427750 + 1.67840i −0.500000 − 0.866025i −0.0733879 − 0.127112i 1.66741 + 0.468758i −1.40439 + 2.43247i −1.00000 −2.63406 + 1.43587i −0.146776
709.5 0.500000 − 0.866025i 0.948586 + 1.44920i −0.500000 − 0.866025i −2.15423 − 3.73123i 1.72934 − 0.0968986i 1.22187 − 2.11634i −1.00000 −1.20037 + 2.74938i −4.30846
709.6 0.500000 − 0.866025i 1.43137 − 0.975290i −0.500000 − 0.866025i 1.26238 + 2.18651i −0.128943 − 1.72724i −2.15398 + 3.73080i −1.00000 1.09762 − 2.79199i 2.52477
709.7 0.500000 − 0.866025i 1.72837 + 0.112872i −0.500000 − 0.866025i 0.109436 + 0.189548i 0.961934 − 1.44038i 0.0782174 − 0.135476i −1.00000 2.97452 + 0.390169i 0.218871
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 355.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1062.2.e.d ✓ 14
9.c even 3 1 inner 1062.2.e.d ✓ 14
9.c even 3 1 9558.2.a.be 7
9.d odd 6 1 9558.2.a.bf 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1062.2.e.d ✓ 14 1.a even 1 1 trivial
1062.2.e.d ✓ 14 9.c even 3 1 inner
9558.2.a.be 7 9.c even 3 1
9558.2.a.bf 7 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1062, [\chi])\):

\( T_{5}^{14} - 4 T_{5}^{13} + 28 T_{5}^{12} - 104 T_{5}^{11} + 554 T_{5}^{10} - 1724 T_{5}^{9} + 4650 T_{5}^{8} + \cdots + 1 \) Copy content Toggle raw display
\( T_{7}^{14} + 21 T_{7}^{12} - 42 T_{7}^{11} + 348 T_{7}^{10} - 606 T_{7}^{9} + 2556 T_{7}^{8} - 4986 T_{7}^{7} + \cdots + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{14} - T^{13} + \cdots + 2187 \) Copy content Toggle raw display
$5$ \( T^{14} - 4 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{14} + 21 T^{12} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( T^{14} + 7 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{14} - 9 T^{13} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( (T^{7} - 14 T^{6} + \cdots - 353)^{2} \) Copy content Toggle raw display
$19$ \( (T^{7} + 11 T^{6} + \cdots + 1877)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 17 T^{13} + \cdots + 71588521 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 2780769289 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 20978915281 \) Copy content Toggle raw display
$37$ \( (T^{7} + 19 T^{6} + \cdots + 310903)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + 23 T^{13} + \cdots + 12924025 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 5360340888081 \) Copy content Toggle raw display
$47$ \( T^{14} - T^{13} + \cdots + 2920681 \) Copy content Toggle raw display
$53$ \( (T^{7} - 36 T^{6} + \cdots + 49977)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$61$ \( T^{14} - 2 T^{13} + \cdots + 201601 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 505400137225 \) Copy content Toggle raw display
$71$ \( (T^{7} - 30 T^{6} + \cdots + 81)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} + 5 T^{6} + \cdots - 2829349)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 21294634681321 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 6118524841 \) Copy content Toggle raw display
$89$ \( (T^{7} + 15 T^{6} + \cdots - 218835)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 122036209 \) Copy content Toggle raw display
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